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Sergey Denisov

Publications and source records attributed to Sergey Denisov.

At least 37 records · Page 2Linked to original sources
arXiv2019

Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Let $μ$ be a measure from Szegő class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $μ$. In this paper, we prove a version of the classical Szegő's formula which controls the oscillation of $f_n$ on $\mathbb T$ for all $n \ge 0$. Then, we focus on an analog of Lusin's conjecture for polynomials $\{φ_n\}$ orthogonal with respect to measure $μ$ and prove that pointwise convergence of $\{|φ_n|\}$ almost everywhere on $\mathbb T$ is equivalent to a certain condition on zeroes of $φ_n$.

math.CV↗
arXiv2019

De Branges canonical systems with finite logarithmic integral

Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a spectral version of the classical Szego theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein-Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

math.CA↗
arXiv2019

Is there a Floquet Lindbladian?

The stroboscopic evolution of a time-periodically driven isolated quantum system can always be described by an effective time-independent Hamiltonian. Whether this concept can be generalized to open Floquet systems, described by a Markovian master equation with time-periodic Lindbladian generator, remains an open question. By using a two level system as a model, we explicitly show the existence of two well-defined parameter regions. In one region the stroboscopic evolution can be described by a Markovian master equation with a time-independent Floquet Lindbladian. In the other it cannot; but here the one-cycle evolution operator can be reproduced with an effective non-Markovian master equation that is homogeneous but non-local in time. Interestingly, we find that the boundary between the phases depends on when the evolution is stroboscopically monitored. This reveals the non-trivial role played by the micromotion in the dynamics of open Floquet systems.

quant-ph↗
arXiv2019

Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes

We consider transport in two billiard models, the infinite horizon Lorentz gas and the stadium channel, presenting analytical results for the spreading packet of particles. We first obtain the cumulative distribution function of traveling times between collisions, which exhibits non-analytical behavior. Using a renewal assumption and the Lévy walk model, we obtain the particles' probability density. For the Lorentz gas, it shows a distinguished difference when compared with the known Gaussian propagator, as the latter is valid only for extremely long times. In particular, we show plumes of particles spreading along the infinite corridors, creating power-law tails of the density. We demonstrate the slow convergence rate via summation of independent identically distributed random variables on the border between Lévy and Gauss laws. The renewal assumption works well for the Lorentz gas with intermediately sized scattering centers, but fails for the stadium channel due to strong temporal correlations. Our analytical results are supported with numerical samplings.

cond-mat.stat-mech↗
arXiv2019

Propagating large open quantum systems towards their steady states: cluster implementation of the time-evolving block decimation scheme

Many-body quantum systems are subjected to the Curse of Dimensionality: The dimension of the Hilbert space $\mathcal{H}$, where these systems live in, grows exponentially with systems' 'size' (number of their components, "bodies"). It means that, in order to specify a state of a quantum system, we need a description whose length grows exponentially with the system size. However, with some systems it is possible to escape the curse by using low-rank tensor approximations known as `matrix-product state/operator (MPS/O) representation' in the quantum community and `tensor-train decomposition' among applied mathematicians. Motivated by recent advances in computational quantum physics, we consider chains of $N$ spins coupled by nearest-neighbor interactions. The spins are subjected to an action coming from the environment. Spatially disordered interaction and environment-induced decoherence drive systems into non-trivial asymptotic states. The dissipative evolution is modeled with a Markovian master equation in the Lindblad form. By implementing the MPO technique and propagating system states with the time-evolving block decimation (TEBD) scheme (which allows to keep the length of the state descriptions fixed), it is in principle possible to reach the corresponding steady states. We propose and realize a cluster implementation of this idea. The implementation on four nodes allowed us to resolve steady states of the model systems with $N = 128$ spins.

cond-mat.stat-mech↗
arXiv2019

Universal spectra of random Lindblad operators

To understand typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markovian completely positive evolution in the space of density matrices. Spectral properties of these operators, including the shape of the spectrum in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate universality of the spectral features. The notion of ensemble of random generators of Markovian qauntum evolution constitutes a step towards categorization of dissipative quantum chaos.

quant-ph↗
arXiv2019

Asymptotic Floquet states of open quantum systems: The role of interaction

We investigate the asymptotic state of a periodically driven many-body quantum system which is weakly coupled to an environment. The combined action of the modulations and the environment steers the system towards a state being characterized by a time-periodic density operator. To resolve this asymptotic non-equilibrium state at stroboscopic instants of time, we introduce the dissipative Floquet map, evaluate the stroboscopic density operator as its eigen-element and elucidate how particle interactions affect properties of the density operator. We illustrate the idea with a periodically modulated Bose-Hubbard dimer and discuss the relations between the interaction-induced bifurcations in a mean-field dynamics and changes in the characteristics of the genuine quantum many-body state. We argue that Floquet maps provide insight into the system relaxation towards its asymptotic state and may help to understand whether it is possible (or not) to construct a stroboscopic time-independent generator mimicking the action of the original time-dependent one.

cond-mat.quant-gas↗
arXiv2018

On the Essential Spectrum of Schrödinger Operators on Trees

It is known that the essential spectrum of a Schrödinger operator $H$ on $\ell^{2}\left(\mathbb{N}\right)$ is equal to the union of the spectra of right limits of $H$. The natural generalization of this relation to $\mathbb{Z}^{n}$ is known to hold as well. In this paper we generalize the notion of right limits to general infinite connected graphs and construct examples of graphs for which the essential spectrum of the Laplacian is strictly bigger than the union of the spectra of its right limits. As these right limits are trees, this result is complemented by the fact that the equality still holds for general bounded operators on regular trees. We prove this and characterize the essential spectrum in the spherically symmetric case.

math.SP↗
arXiv2017

Asymptotic Floquet states of non-Markovian systems

We propose a method to find asymptotic states of a class of periodically modulated open systems which are outside the range of validity of the Floquet theory due to the presence of memory effects. The method is based on a Floquet treatment of the time-local, memoryless dynamics taking place in a minimally enlarged state space where the original system is coupled to auxiliary -- typically non-physical -- variables. A projection of the Floquet solution into the physical subspace returns the sought asymptotic state of the system. The spectral gap of the Floquet propagator acting in the enlarged state space can be used to estimate the relaxation time. We illustrate the method with a modulated version of quantum random walk model.

cond-mat.stat-mech↗
arXiv2017

Limit theorems for Lévy walks in $d$ dimensions: rare and bulk fluctuations

We consider super-diffusive Lévy walks in $d \geqslant 2$ dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the $d-$dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic $d-$dimensional Lévy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the $d-$dimensional process can be reduced to one-dimensional Lévy walk.

cond-mat.stat-mech↗
arXiv2016

Effective Floquet-Gibbs states for dissipative quantum systems

A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet-Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diagonal elements obeying a Gibbs distribution, being parametrized by the corresponding Floquet quasi-energies. Addressing the non-adiabatic driving regime, upon using the Magnus expansion, we employ the concept of a corresponding effective Floquet Hamiltonian. In doing so we go beyond the conventionally used RWA and demonstrate that the idea of Floquet-Gibbs states can be extended to the realistic case of a weak, although finite system-bath coupling, herein termed effective Floquet-Gibbs states.

cond-mat.stat-mech↗
arXiv2016

Experimental control of transport resonances in a coherent quantum rocking ratchet

The ratchet phenomenon is a means to get directed transport without net forces. Originally conceived to rectify stochastic motion and describe operational principles of biological motors, the ratchet effect can be used to achieve controllable coherent quantum transport. This transport is an ingredient of several perspective quantum devices including atomic chips. Here we examine coherent transport of ultra-cold atoms in a rocking quantum ratchet. This is realized by loading a rubidium atomic Bose-Einstein condensate into a periodic optical potential subjected to a biharmonic temporal drive. The achieved long-time coherence allows us to resolve resonance enhancement of the atom transport induced by avoided crossings in the Floquet spectrum of the system. By tuning the strength of the temporal modulations, we observe a bifurcation of a single resonance into a doublet. Our measurements reveal the role of interactions among Floquet eigenstates for quantum ratchet transport.

cond-mat.quant-gas↗
arXiv2015

Seasonal Floquet states in a game-driven evolutionary dynamics

Mating preferences of many biological species are not constant but season-dependent. Within the framework of evolutionary game theory this can be modeled with two finite opposite-sex populations playing against each other following the rules that are periodically changing. By combining Floquet theory and the concept of quasi-stationary distributions, we reveal existence of metastable time-periodic states in the evolution of finite game-driven populations. The evolutionary Floquet states correspond to time-periodic probability flows in the strategy space which cannot be resolved within the mean-field framework. The lifetime of metastable Floquet states increases with the size $N$ of populations so that they become attractors in the limit $N \rightarrow \infty$.

q-bio.PE↗
arXiv2014

Infinite densities for Lévy walks

Motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion. In many cases, such systems exhibit strong anomalous diffusion, where low order moments $< |x(t)|^q >$ with $q$ below a critical value $q_c$ exhibit diffusive scaling while for $q>q_c$ a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent $α$ and the diffusion coefficient $K_α$. We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

cond-mat.stat-mech↗